Eigenvalues Of Certain Matrices Research Articles

On the extreme eigenvalues of certain matrices of non-standard inner products of Hermite polynomials

We study Hermite orthogonal polynomials and Gram matrices of their non-standard inner products. The weight function of the non-standard inner product is obtained from the Gauss probability density function by its horizontal shift by a real parameter. We are interested in the spectral properties of these matrices and some of their modifications. We show how the largest and smallest eigenvalues of the matrices depend on the parameter.

The Montecinos–Balsara ADER-FV polynomial basis: Convergence properties & extension to non-conservative multidimensional systems

Hyperbolic systems of PDEs can be solved to arbitrary orders of accuracy by using the ADER Finite Volume method. These PDE systems may be non-conservative and non-homogeneous, and contain stiff source terms. ADER-FV requires a spatio-temporal polynomial reconstruction of the data in each spacetime cell, at each time step. This reconstruction is obtained as the root of a nonlinear system, resulting from the use of a Galerkin method. It was proved in [7] that for traditional choices of basis polynomials, the eigenvalues of certain matrices appearing in these nonlinear systems are always 0, regardless of the number of spatial dimensions of the PDEs or the chosen order of accuracy of the ADER-FV method. This guarantees fast convergence to the Galerkin root for certain classes of PDEs.In Montecinos and Balsara [8] a new, more efficient class of basis polynomials for the one-dimensional ADER-FV method was presented. This new class of basis polynomials, originally presented for conservative systems, is extended to multidimensional, non-conservative systems here, and the corresponding property regarding the eigenvalues of the Galerkin matrices is proved.

Covariance Inflation in the Ensemble Kalman Filter: A Residual Nudging Perspective and Some Implications

Abstract This article examines the influence of covariance inflation on the distance between the measured observation and the simulated (or predicted) observation with respect to the state estimate. In order for the aforementioned distance to be bounded in a certain interval, some sufficient conditions are derived, indicating that the covariance inflation factor should be bounded in a certain interval, and that the inflation bounds are related to the maximum and minimum eigenvalues of certain matrices. Implications of these analytic results are discussed, and a numerical experiment is presented to verify the validity of the analysis conducted.

Removable sets and approximation of eigenvalues and eigenfunctions on combinatorial graphs

A new concept of a removable set of vertices on a combinatorial graph is introduced. It is shown that eigenfunctions of a combinatorial Laplace operator L on a graph G which correspond to small eigenvalues can be reconstructed as limits of the so-called variational splines. Spaces of such variational splines determined by uniqueness sets which are compliments of removable sets. It is important that in some cases this spaces have very small dimension and can be described explicitly. It is shown that small eigenvalues of L can be approximated by eigenvalues of certain matrices acting in spline spaces. The results have potential applications to various problems such as high-dimensional data dimension reduction, image processing, computer graphics, visualization and learning theory.

Extreme rays for the large cone of hermitian generalized matrix functions

Given a <artwork name="GLMA31007ei1">-valued function f with domain <artwork name="GLMA31007ei2">, the symmetric group on {1,2,…, m}, we define the generalized matrix function [ f ](⋅), or df (⋅), in the usual way on the set of all m× m complex matrices. Letting <artwork name="GLMA31007ei3"> denote the set of all m× m positive semi-definite Hermitian matrices we consider the cone K m whose elements are the Hermitian functions <artwork name="GLMA31007ei4"> such that [ f ]( A)≥0 for all <artwork name="GLMA31007ei5">. The extreme rays in K m are fundamental to an understanding of the linear inequalities that result by restricting the generalized matrix functions [ f ](⋅) to the sets <artwork name="GLMA31007ei6">. In particular, the resolution of Lieb's permanent dominance conjecture, and certain similar conjectures such as the conjecture of Soules, will likely require identification and careful analysis of these rays. Grone, Merris, and Watkins have shown that the determinant function det(⋅), which is [ f ](⋅) if f is the signum function, is extreme in K m for each m. We identify additional rays that are extreme for all m. In particular, we associate with each 2-term partition <artwork name="GLMA31007ei7"> of {1,2,…, m} an element <artwork name="GLMA31007ei8"> that is shown to be extreme in K m for each m. If <artwork name="GLMA31007ei9"> is trivial, then <artwork name="GLMA31007ei10"> reduces to the determinant function; hence, our results are a natural extension of the result of Grone, Merris, and Watkins. Moreover <artwork name="GLMA31007ei11">, like det ( A), is expressible as a function of the eigenvalues of certain matrices related to A. Additional classes of extreme rays are also presented.

Inequalities for hadamard product and unitarily invariant norms of matrices

The paper contains some general theorems for Hadamard product of matrices which in particular include Fiedler's Theorem and a better bound for an inequality on product of eigenvalues of certain matrices due to Ando. Lieb's concavity Theorem has been proved using operator means. Some inequalities for unitarily invariant norms have also been proved.

Hamilton cycles and eigenvalues of graphs

We prove some results concerning necessary conditions for a graph to be Hamiltonian in terms of eigenvalues of certain matrices associated with the graph. As an example, we show how the results give an easy algebraic proof of the nonexistence of a Hamilton cycle in two graphs, one of them being the Petersen graph.

On the Convergence of Reversible Markov Chains

A simple upper bound for the second-largest eigenvalue of a finite reversible time-homogeneous Markov chain is presented as a function of the transition probabilities, the equilibrium distribution, and the underlying structure of the chain. This work is extended to a relation between the second-largest eigenvalues of any two reversible Markov chains with the same underlying structure. Furthermore, a lower bound for the smallest eigenvalue of a reversible chain is also presented, thereby providing a bound on the spectral gap of such chains. These eigenvalue bounds are fairly easy to compute for a variety of reversible chains by using known results on eigenvalues of certain matrices associated with graphs or random walks on graphs. The results on the spectral gap lead to a bound on the time constant of a reversible Markov chain converging to its equilibrium distribution. As an application, the temperature asymptotics of simulated annealing, which is a probabilistic algorithm widely used for solving combinat...

The preconditioned simultaneous displacement method (PSD method) for elliptic difference equations

This paper introduces the Preconditioned Simultaneous Displacement iterative method (PSD method) in a new “computable” form for the numerical solution of linear systems of the form Au=b, where the matrix A is large and sparse. The convergence properties of the method are analysed under certain assumptions on the matrix A. Moreover, “good” values (near the optimum) for the involved parameters are determined in terms of bounds on the eigenvalues of certain matrices. Bounds on the reciprocal rate of convergence of the PSD method are also given. The method is shown to be superior over the well known Symmetric Successive Overrelaxation method (SSOR method) (at the optimum stage PSD is shown to converge approximately two times faster than SSOR) and in certain cases over the Successive Overrelaxation method (SOR method).

A Monte Carlo study of local identifiability and degrees of freedom in the asymptotic likelihood ratio test

In applications to restricted factor analysis, it is shown that the determination of local identifiability and degrees of freedom in the asymptotic likelihood ratio test, from the eigenvalues of certain matrices, is not unambiguous in all cases. Some results are given that serve to indicate appropriate reparameterizations of unidentified problems.

An eigenvalue characterization of multivariable system zeros

A characterization of multivariable system zeros as the common eigenvalues of certain matrices is obtained. Major differences are noted between this and other approaches, including implementation and interpretation of results. In addition, a useful determinantal identity is proven.

A lower bound on the eigenvalues of certain hamiltonian matrices

A lower bound is obtained for the eigenvalue of certain matrices arising from the application of the theory of the symmetric groups to the calculation of energy for n-electron systems.